Elliptic Quaternions

TauQuaternion = TauReal-algebra {1,i,j,k} with Hamilton product: i^2=j^2=k^2=ijk=-1. Non-commutativity is earned as a structural consequence of extending beyond two dimensions.

Elliptic Quaternions

Summary

TauQuaternion = TauReal-algebra {1,i,j,k} with Hamilton product: i^2=j^2=k^2=ijk=-1. Non-commutativity is earned as a structural consequence of extending beyond two dimensions.

Statement

%
\label{def:tau-quaternion}
The \textbf{$\tau$-quaternion algebra} is the $\mathbb{R}_\tau$-algebra:
\[
    \boxed{%
    \mathbb{H}_\tau
    := \mathbb{R}_\tau \langle i, \mathbf{j}, k \rangle
    \;/\;
    (i^2 = \mathbf{j}^2 = k^2 = i\mathbf{j}k = -1),}
\]
with elements written as $(a, b, c, d) \in \mathbb{R}_\tau^4$,
representing $q = a + bi + c\mathbf{j} + dk$.
The multiplication is determined by the cyclic rules
$ij = k$, $jk = i$, $ki = j$ (positive cycle)
and $ji = -k$, $kj = -i$, $ik = -j$ (negative cycle),
extended bilinearly to all of $\mathbb{H}_\tau$.

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-01.jsonl line 193
• Manuscript source: 2nd-edition/book-i-categorical-foundations/02_mainmatter/part17/ch78-elliptic-quaternions.tex lines 40-57

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookI.Boundary.Quaternions
• Name: Tau.Boundary.TauQuaternion

Dependencies

• Canonical: I.D84

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.